This paper shows that system throughput and the number of PACs generated by a given time in a PAC system are jointly concave with respect to the initial inventory and the number of process cards at each state. The result is easily proved by observing that with exponential service times (and general arrival process), the (counting) job completion, job departure, and card generation processes are increasing concave functions of the initial inventory and card counts. The result applies to a wide range of multi-stage production-inventory systems including the classical base stock policy, integral control and the kanban system, and is crucial for the design of such systems.
